The Senate’s Gang of Eight is mounting an aggressive lobbying campaign with one major goal: weaken the conservative opposition to a sweeping immigration overhaul.

Hours after the bill was unveiled after 2 a.m. Wednesday, Republicans began to make their pitch for it on conservative talk radio and by urging their colleagues to hold their fire until they’ve had a chance to analyze it. Their hope is to define the measure as a workable compromise on a highly complex issue, one that will help broaden the GOP’s reach to Latino and moderate voters.

The push is part of a broader strategy to smooth passage for the complex legislation in the Senate, where the idea is to lure more than just a handful of Republican senators. If a broadly backed bill passed the Senate, House Republicans would be hard pressed to reject it, proponents believe.

“Defining the bill quickly is good,” said Sen. Lindsey Graham (R-S.C.), a member of the gang, in an interview. “We’re going to be aggressive in marketing the bill. We’re going to be open minded about making it better. But this is an all-hands-on-deck approach.”

On Wednesday, Graham and Sen. Marco Rubio made the rounds on conservative talk radio, with the Florida Republican planning for more appearances on Thursday. The initial reaction on the right was far less intense than they anticipated, proponents said. Rubio’s offices in Washington and Florida received fewer than 500 calls on immigration Wednesday, with more than 150 of those in favor of his bill, according to a source in his office.

And some GOP senators were circumspect, saying they would wait and see the details of the bill before making their minds up whether to pick it apart.

“I’ll take another look at it,” said Sen. Thad Cochran (R-Miss.), who voted to block the 2007 bill.

“Is that enough for me? I don’t want to say yet. But it was a lot better than I thought it was,” said Sen. Chuck Grassley, the top Republican on the Senate Judiciary Committee, about the border security measures.

On the left, Democrats tried to shore up their own base. Sen. Chuck Schumer (D-N.Y.), another leading supporter, and Democratic senators privately met with a room full of immigration activists to implore them to put aside their fears for the “greater good” of passing the bill. And a senior White House official, Cecilia Muñoz, met with House Democrats in House Minority Whip Steny Hoyer’s office, offering broad support for the bill, hearing from Democrats worried about elements of the bill’s treatment of undocumented immigrants. The White House would also like the bill tweaked.

The public-private campaign marked the beginning of a furious lobbying push by Republicans to limit the outcry from the right over “amnesty” for illegal immigrants and Democrats to quiet dissension from the left over the proposal’s treatment of 11 million people living in the country without the proper papers. Both sides believe that if they limit knee-jerk outcry at the start of the process, it will make it harder for the opposition to mobilize.

“I think our Republican colleagues are torn: A good portion of their right-wing constituency is against the bill,” Schumer said. “But at the same time, they know that the way the Republican Party can get well nationally is by getting something done. And so if the House members, which are more conservative, see only five or six [Senate] Republicans voting for this, it’s a whale of a difference than if we see 20.”

But it didn’t take long for the senators to see they still have their work cut out for them.

House Republicans, like Rep. Lamar Smith of Texas, were already bashing the bill, and a Senate GOP leader, John Cornyn, was wary about the border security provisions in the proposal. A prospective 2016 presidential candidate, Sen. Rand Paul of Kentucky, was preparing to offer amendments designed to pull the bill to the right. And pro-immigration activist groups were eager to move it leftward, including working to ensure that same-sex couples are afforded protections now ignored in the massive proposal.

“It’s worse than we thought,” said Smith, who formerly chaired the Judiciary Committee. He added: “It’s amnesty on a massive scale, greater than we anticipated,” Smith said. “And we took their word that the border was going to be secured before the other reforms were implemented and that’s not the case.”

“As smart as these eight senators are, they are not from Texas,” Cornyn said.

If enacted, the bill would make the biggest changes to immigration laws since 1986, when a statute was enacted that is frequently bashed by Republicans as giving amnesty to illegal immigrants. But proponents of the 2013 measure argue that it’s nothing like the 1986 law — or the 2007 proposal, for that matter — saying there are enormous differences over border security, new electronic verification requirements for employers and new visa programs to attract foreign low-skilled and high-skilled workers into the country.

To appease conservative concerns over legalizing undocumented immigrants, the bill would call for billions of dollars to be spent on tightened security at the U.S.-Mexico border with a goal of apprehending 90 percent of those crossing the border in “high-risk” areas. Immigrants would pay a total of $2,000 in fines, pass a background check, have a job and wait 10 years before applying for a green card. Three years after that, they could apply to become U.S. citizens. But the whole process is contingent on the government meeting a series of border-security benchmarks.

As House Republicans privately discuss immigration, the more leadership realizes it’s an uphill climb. Most Republicans are not focused on the pathway to citizenship, but rather border security. A bipartisan group of House members is working on its own proposal but is stalled and has not released either principles or legislative text.

The House Judiciary Committee will begin moving small-bore immigration bills in the coming days.

The complaints aren’t limited to the substance of the bill, either. Rep. Jason Chaffetz (R-Utah), a member of the Judiciary Committee, said the comprehensive approach that the Senate took won’t fly in the House.

“We need to do this incrementally,” Chaffetz said in an interview. “If it’s a big bill, it’ll die under its own weight because there will be something for everybody to hate. All or nothing is a losing strategy.”

House Democrats weren’t so quick to embrace the Senate bill either. They say they are tired of having no say in the legislative process, and are beginning to push back.

“The House has got to have its say in these matters,” Rep. John Larson (D-Conn.) said. “We can’t continue to operate exclusively as a messaging body, and only the body that has the last recourse court on the way to passage.”

Democrats watched President Barack Obama’s efforts to pass gun restrictions fail in the Senate and said it might foreshadow what will happen on immigration reform.

“It’s like guns,” Rep. Bill Pascrell (D-N.J.) said. “Everyone thought everything looks wonderful, but this is a long way to go yet.”

Several Senate Republicans who voted against the comprehensive bill in 2007 are waiting for the legislative process and the politics to play out before taking a position, a positive development for the negotiators who are trying to build support for the plan before the opposition takes shape.

“I’m very open-minded,” said Sen. Orrin Hatch (R-Utah). “People want to do what’s right here, but there are lot of concerns on what is the best way to handle this problem and not create more.”